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In a locality, two-thirds of the people have cable TV, one- fifth have VCR, and one-tenth have both. What is the fraction of people having either cable -TV or VCR?

  1. $19/30$
  2. $2/3$
  3. $17/30$
  4. $23/30$
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Given that,

  • $n(\text{Cable TV)}=\frac{2}{3}$
  • $n(\text{VCR})=\frac{1}{5}$
  • $n(\text{Cable TV $\cap$ VCR})=\frac{1}{10}$

Lets draw the venn diagram.

Now, the fraction of people having either Cable TV or VCR

$= n(\text{Cable TV $\cup$ VCR})-n(\text{Cable TV $\cap$ VCR})$

$= n(\text{Cable TV)+n(VCR)}-n(\text{Cable TV $\cap$ VCR})-n(\text{Cable TV $\cap$ VCR})$

$=\frac{2}{3}+\frac{1}{5}-\frac{1}{10}-\frac{1}{10}$

$=\frac{2}{3}+\frac{1}{5}-\frac{2}{10}$

$=\frac{2}{3}+\frac{1}{5}-\frac{1}{5}$

$=\frac{2}{3}$

                      (OR)

The fraction of people having either Cable TV or VCR

$= n(\text{Cable TV})-n(\text{Cable TV $\cap$ VCR})+n(\text{VCR})-n(\text{Cable TV $\cap$ VCR})$

$=\frac{2}{3}-\frac{1}{10}+\frac{1}{5}-\frac{1}{10}$

$=\frac{2}{3}$

Correct Answer : $\text{B}$

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